Related papers: Three classes of 1-planar graphs
By a poly-line drawing of a graph G on n vertices we understand a drawing of G in the plane such that each edge is represented by a polygonal arc joining its two respective vertices. We call a turning point of a polygonal arc the bend. We…
In this paper, we discuss matching extendability of optimal $1$-projective plane graphs (abbreviated as O1PPG), which are drawn on the projective plane $P^2$ so that every edge crosses another edge at most once, and has $n$ vertices and…
A shape visibility representation displays a graph so that each vertex is represented by an orthogonal polygon of a particular shape and for each edge there is a horizontal or vertical line of sight between the polygons assigned to its…
We establish a one-to-one correspondence between 1-planar graphs and general and hole-free 4-map graphs and show that 1-planar graphs can be recognized in polynomial time if they are crossing-augmented, fully triangulated, and maximal…
An odd coloring of a graph $G$ is a proper coloring such that any non-isolated vertex in $G$ has a coloring appears odd times on its neighbors. The odd chromatic number, denoted by $\chi_o(G)$, is the minimum number of colors that admits an…
We show that for every integer $n\geq 1$ there exists a graph $G_n$ with $(1+o(1))n$ vertices and $n^{1 + o(1)}$ edges such that every $n$-vertex planar graph is isomorphic to a subgraph of $G_n$. The best previous bound on the number of…
Not every directed acyclic graph (DAG) whose underlying undirected graph is planar admits an upward planar drawing. We are interested in pushing the notion of upward drawings beyond planarity by considering upward $k$-planar drawings of…
A geometric graph is a graph drawn in the plane so that its vertices and edges are represented by points in general position and straight line segments, respectively. A vertex of a geometric graph is called pointed if it lies outside of the…
We show that every graph with two crossings is 5-choosable. We also prove that every graph which can be made planar by removing one edge is 5-choosable.
We settle a problem of Dujmovi\'c, Eppstein, Suderman, and Wood by showing that there exists a function $f$ with the property that every planar graph $G$ with maximum degree $d$ admits a drawing with noncrossing straight-line edges, using…
Every planar graph has a 4-page book embedding and this bound is tight. We show that every 1-planar graph, which is a graph that admits a drawing with at most one crossing per edge, has a 10-page book embedding. In addition, four pages are…
A geometric graph is a graph drawn in the plane with vertices represented by points and edges as straight-line segments. A geometric graph contains a (k,l)-crossing family if there is a pair of edge subsets E_1,E_2 such that |E_1| = k and…
A $(c_1,c_2,...,c_k)$-coloring of $G$ is a mapping $\varphi:V(G)\mapsto\{1,2,...,k\}$ such that for every $i,1 \leq i \leq k$, $G[V_i]$ has maximum degree at most $c_i$, where $G[V_i]$ denotes the subgraph induced by the vertices colored…
The crossing number of a graph $G$ is the least number of crossings over all possible drawings of $G$. We present a structural characterization of graphs with crossing number one.
In a book embedding, the vertices of a graph are placed on the spine of a book and the edges are assigned to pages, so that edges on the same page do not cross. In this paper, we prove that every $1$-planar graph (that is, a graph that can…
A graph is $(c_1, c_2, ..., c_k)$-colorable if the vertex set can be partitioned into $k$ sets $V_1,V_2, ..., V_k$, such that for every $i: 1\leq i\leq k$ the subgraph $G[V_i]$ has maximum degree at most $c_i$. We show that every planar…
We generalize the notions of flippable and simultaneously flippable edges in a triangulation of a set S of points in the plane to so-called \emph{pseudo-simultaneously flippable edges}. Such edges are related to the notion of convex…
Algorithmic extension problems of partial graph representations such as planar graph drawings or geometric intersection representations are of growing interest in topological graph theory and graph drawing. In such an extension problem, we…
A topological graph is \emph{$k$-quasi-planar} if it does not contain $k$ pairwise crossing edges. A topological graph is \emph{simple} if every pair of its edges intersect at most once (either at a vertex or at their intersection). In…
We consider the problem of drawing an outerplanar graph with $n$ vertices with at most one bend per edge if the outer face is already drawn as a simple polygon. We prove that it can be decided in $O(nm)$ time if such a drawing exists, where…